Frances A. Rosamond

On the parameterized complexity of multiple-interval graph problems

Multiple-interval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multiple-interval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multiple-interval graphs was initiated. In this sequel, we study multiple-interval graph problems from the perspective of parameterized complexity. The problems under consideration are k-Independent Set, k-Dominating Set, and k-Clique, which are all known to be W[1]-hard for general graphs, and NP-complete for multiple-interval graphs. We prove that k-Clique is in FPT, while k-Independent Set and k-Dominating Set are both W[1]-hard. We also prove that k-Independent Dominating Set, a hybrid of the two above problems, is also W[1]-hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]-hardness via a reduction from the k-Multicolored Clique problem, a variant of k-Clique. We believe this technique has interest in its own right, as it should help in simplifying W[1]-hardness results which are notoriously hard to construct and technically tedious.

Cutting Up is Hard to Do: the Parameterized Complexity of k-Cut and Related Problems.

Abstract The Graph k-Cut problem is that of finding a set of edges of minimum total weight, in an edge-weighted graph, such that their removal from the graph results in a graph having at least k connected components. An algorithm with a running time of O(nk2) for this problem has been known since 1988, due to Goldschmidt and Hochbaum. We show that the problem is hard for the parameterized complexity class W[1]. We also investigate the complexity of a related problem, C utting A F ew V ertices from a G raph , that asks for the minimum cost of separating at least k vertices from an edge-weighted connected graph. We show that this problem also is hard for W[1].

Clique-Width is NP-Complete

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless

Graph Layout Problems Parameterized by Vertex Cover

P = NP

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

. We also show that, given a graph

Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity

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An O (2 O(k) n 3 ) FPT algorithm for the undirected feedback vertex set problem

and an integer

Coordinatized Kernels and Catalytic Reductions: An Improved FPT Algorithm for Max Leaf Spanning Tree and Other Problems

k

The undirected feedback vertex set problem has a poly( k ) kernel

, deciding whether the clique-width of

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

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